A small marble is suspended from a 45 cm string and creates a conical pendulum (like in the picture beside question #57). The marble revolves at constant speed in a horizontal circle with a 6.5 cm radius. Calculate the marble's velocity along the circle.
To calculate the marble's velocity along the circle, we can use the equation for the speed of an object in circular motion:
v = ω * r
where v is the velocity, ω is the angular velocity, and r is the radius of the circle.
In this case, the radius of the circle is given as 6.5 cm. However, we need to find the angular velocity (ω) of the marble.
To find the angular velocity, we need to determine the period (T) of the circular motion. The period is the time it takes for the marble to complete one full revolution.
To find the period, we can use the length of the string and the acceleration due to gravity:
T = 2π * √(L / g)
where T is the period, L is the length of the string, and g is the acceleration due to gravity.
In this case, the length of the string is given as 45 cm. The acceleration due to gravity, g, is approximately 9.8 m/s².
Converting the length of the string to meters:
L = 0.45 m
Substituting the values into the formula:
T = 2π * √(0.45 / 9.8)
Calculating the period:
T ≈ 2.88 s
Now that we have the period, we can calculate the angular velocity using the formula:
ω = 2π / T
Substituting the value of the period:
ω ≈ 2π / 2.88
Calculating the angular velocity:
ω ≈ 2.18 rad/s
Finally, we can use the angular velocity and the radius of the circle to calculate the velocity along the circular path:
v = 2.18 rad/s * 0.065 m
Calculating the velocity:
v ≈ 0.142 m/s
Therefore, the marble's velocity along the circle is approximately 0.142 m/s.